632 
PROFESSOR SYLVESTER ON THE REAL 
ing two real, according as the sign of cSF(V) is the same as or the contrary to that of 
Accordingly, if r is real ( 45 ) and i even, the nature of the ensemble of the i 
roots r-\-dr will not be the same when ^F(r) is positive as when e>F(r) is negative, 
(50) So, further, if F^=0 have 2 m equal roots r, 2 n equal roots s, and so on, the deduced 
corresponding groups of roots in F(#)+t5F(^):=0 will, or may at least each of them, 
undergo a change of character to the extent of one pair of the r group changing their 
nature with the sign of c$F(r), one pair of the s group changing their nature with the 
sign of &F(s), and so on ; but in no case, except F(#) possess some equal roots (i. e. 
unless its discriminant be zero) , can an infinitesimal variation in the constants affect the 
character of the roots ( 46 ). 
(51) To every facultative point corresponds a certain set of values of J, D, L; and when 
these are given, it has been shown that the equation (a, b, c, d, e,fjx, yf is reducible to 
the form nt 5 +st; 5 -Hw 5 , where w+v+w=0, or to the form ru]-\-sv]-\-tw 5 , where 
, , _ i — w + iv —w—iv 
w ; +v ; +w=0, and u t — — ^ — > v ,= — ^ — 5 
or to the form au 5 -\-beuv 4 u, v, w being always real linear functions of x, y, with 
the sole exception that when J=0, K=0, L=0, the reduced form is 
a u 5 -J- 5 bu 4 v + 1 0 cv?y 2 . 
When these three invariants are not all zero, the coefficients in the reduced form r , s, t 
or a, e, f are known functions of J, D, L, and the character of the roots is perfectly deter- 
minate ; so that to every facultative point corresponds an infinite family of equations 
with real linear coefficients all deducible from each other by real linear substitutions. 
Thus then, with the sole exception of the origin, every facultative point corresponds to 
a determinate character of equation, viz. to an equation with four, or two, or no imagi- 
nary roots ; so that by a bold figure of speech we may be permitted to speak of every 
point but one in facultative space having a determinate quality, as masculine, feminine, 
or neuter. The origin alone is exempt from this law, and may be considered to be of 
epicene gender, since the factor £m 2 -j-55m;+10t; 2 may have its roots real or imaginary. 
As we travel continuously from point to point in the facultative portion of space we 
pass from family to family, or, if we please, from an individual of one family to an indi- 
vidual of another family, differing from the former individual by an infinitesimal varia- 
tion of the constants. 
( 45 ) r, although, supposed to he one of a group of equal roots, is not necessarily real, for it may belong to a 
factor (a? + 2e cos 0 + e 2 ) 2 . 
( 46 ) Compare this statement with the corresponding one given by M. Hekmite, Camb. and Dub. Journal, vol. ix. 
p. 204, where only one parameter is supposed to undergo a change. I think that greater breadth and at the 
same time greater precision and clearness are gained by the mode of exposition employed in the text above. It 
will be observed that for a change of character to he possible when the function passes through a phase of equal 
roots, it is not enough that there shall exist a group of equal roots r, hut there must be an even number of 
such roots in the group, and, furthermore, the equal roots must he real ; when this last supposition is, not 
satisfied, no change in the character of dr will affect the character of r-\-dr: an instructive exemplification of 
this remark will occur in the sequel. 
